Over the last few years, finite element algorithms for solution of the Euler flow equations have gained increased popularity. The current work is considered with a new way to solve the flow equations using a variational technique. A computer code named FLUIDEX was developed to analyze barotropic fluid dynamics. The solution of the flow problem is obtained by using numerical algorithm to find the extremum value of an "Action" i.e. by a variational principle. Predictions of the FLUIDEX numerical model were analyzed for particular cases of potential flow (compressible and incompressible). The results were successfully compared against exact analytical solutions for potential flow test problems and for approximate linearized analytical solution for compressible irorational flow at low Mach numbers. The proposed method obtains fast and stable solutions without the need to integrate the equations in time and space, and thus enables a considerable reduction of the time and cost of the solution.